Why isn't the reverse KL divergence commonly used in supervised learning?
It’s not an exhaustive answer to your question, but here some aspects that might be helpful:

A common problem in supervised learning where you will see KL-divergence used are classification tasks. Very often in those cases the data points in the training set are assumed to belong to a single class $c_i$ where $i$ is from some index set $I$. The class membership is represented by one-hot encoding, which corresponds to a distribution $P$ with $p_i = 1$ for the class the data point belongs to and $p_j = 0$ for all $i \neq j$. The loss function mostly used in this problems is the average categorical cross-entropy $\langle E_P[-log(Q)]\rangle_{\rm data} = \langle H(P) - D(P || Q)\rangle_{\rm data}$, which in the case of single-class membership reduces to the binary cross-entropy. Because only one probability in $P$ is non-zero in this case, the cross-entropy simplifies to $\langle -\log(q_i)\rangle_{\rm data}$. The reverse KL-Divergence $D(Q || P)$ is strictly speaking not even defined in this case, as $Q$ needs to be absolute continuous w.r.t. $P$. 

There are of course cases were class membership is probabilistic also in the training data and the above argument doesn't apply in full. From a more conceptual point of view, this thread summaries nicely the relation of cross-entropy to maximizing the likelihood of the observed data under the model. Using the reverse cross-entropy would maximize the likelihood of a typical sample drawn from the model under the (empirical) distribution of the observed data. 

A small side note when you read the above link is that in the above described classification problem we have a probability model for the class membership for each point in the input space, this is why the average cross-entropy is minimized.

Classification problems are just one example of supervised learning, so this answer might not fully cover your question.